We study the spectral properties of a rank-one multiplicative perturbation of a unitary matrix, a model introduced by Fyodorov. Building upon earlier results by Forrester and Ipsen, we provide a direct proof that the eigenvalues converge to the zeros of a specific Gaussian analytic function. Our approach extends these results to other unitarily invariant models. This method enables us to address a question raised by Dubach and Reker concerning the critical timescale at which an outlier emerges.